“Legally” you are allowed to run a charging cable across a pavement to charge your car. However I use this term loosely!
That said the Highways act 1980 however, makes it illegal to place a cable over, along or across a highway or pavement without consent from the local Highway Authority. The local Highways department can issue fines or tell you to remove it, unless the is evidence that all necessary means have been taken to give adequate warning. If you choose to do so, you are fully liable for any injuries, that may arise should somebody trip over the cable.
You can submit an application to your local Highways dept to lay a cable across a pavement temporary basis using something like D Lines cable solution for on stree charging or by using permanent solution, such as Kerbocharge.
For a permanent solution like Kerbocharge, the local Highways dept will most likely instruct the works and own the cable crossing but, issue use of rights license. Kerbocharge costs £1,000 unless you LA has any grants towards the costs. If the pavement isn’t owned by Highways you will likely require planning pernmission.
Its great wanting all the tech an EV might bring but, unless you have the rest to go with it, it might not be the best way to go in the longterm. A 3 pin plug granny charger is slow at charging an electric car, public charging or standard electricity tariff can cost more than running a petrol or hybrid. Having a two rate tariff where electric is cheaper over night (however, a trailing cable when its dark is easier for someone to trip over!) and a home charger ready for when your able to go further a field.
Unless you can meet that a hybrid might be the better way to go imo 🙂